1. **State the problem:** Solve the trigonometric equation $$\tan\left(\frac{\pi}{5}x\right) - 1 = 0$$ for $$0 \leq x \leq 8$$. 2. **Rewrite the equation:** We want to find $$x$$ such that $$\tan\left(\frac{\pi}{5}x\right) = 1$$. 3. **Recall the tangent function property:** $$\tan(\theta) = 1$$ at $$\theta = \frac{\pi}{4} + k\pi$$ for any integer $$k$$. 4. **Set the argument equal to these values:** $$\frac{\pi}{5}x = \frac{\pi}{4} + k\pi$$ 5. **Solve for $$x$$:** $$x = \frac{5}{\pi} \left( \frac{\pi}{4} + k\pi \right) = \frac{5}{4} + 5k$$ 6. **Find all $$x$$ in the interval $$0 \leq x \leq 8$$:** - For $$k=0$$: $$x = \frac{5}{4} = 1.25$$ - For $$k=1$$: $$x = \frac{5}{4} + 5 = 6.25$$ - For $$k=2$$: $$x = \frac{5}{4} + 10 = 11.25$$ (outside interval) - For $$k=-1$$: $$x = \frac{5}{4} - 5 = -3.75$$ (outside interval) 7. **Final solutions:** $$x = 1.25$$ and $$x = 6.25$$. **Answer:** $$\boxed{\{1.25, 6.25\}}$$